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Neutron Transport Theory and Reactor Kinetics

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Neutron Transport Theory and Reactor Kinetics Powered By Docstoc
					Chain Reaction Multiplication Criticality Conditions

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T8: Chain Reaction

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Overview
• • • • • • • Chain Reaction Conversion and Breeding Doubling Time Neutron Cycle in a Thermal Reactor Multiplication Factor k Simplified Fuel Cycle Fuel at Discharge

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n

Chain Reaction
ν

β
235 92

235 92

U

U
ν γ

n  0.1 eV

γ β

n  2 MeV

235 92

U

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Uranium Fuel
238U 235U 234U

4.5 109 yr Unat UE DU (UD) 99.27% 98-95% 99.8%

0.7 109 yr 0.72% 2-5% 0.2%

247000 yr 0.006% 0.006% 0.006%

Application CANDU (D2O) LWR (H2O) Fast Reactor (FR, FBR) Armour for tanks Armour-piercing

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Conversion or Breeding
Fertile: n  238 U  92
239 92

U

n  232 Th  90

233 90

Th  
22.3min

23.5min
239 93

Np    
2.35 d


233 91

Pa     
27.0 d

Fissile:

239 94

Pu   
24110 yr

233 92

U    
159200 yr

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Conversion
10 4
4 1 0

238
10 3

U
1 0
3

2 3 8

U

(barns)

capture
10 1

(barns)

10 2

2 1 0

t o t a l

1 1 0

10 0

0 1 0

c a p t u r e
10 -1
1 1 0

fission
10
-2

10 -3

10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

10 5

10 6

10 7

2 1 0 - 3 1 2 10 1 2 3 4 5 6 7 0 1 0 10 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0

10 4

Energy (eV)

235
10
3

10 4

E) n e rV g y ( e

U
10
3

fission

239

Pu

(barns)

(barns)

10 2

10 2

capture

fission
10 1

10 1

10 0

10 0

capture
10
-1

10 -1

10 -2 -3 10

10

-2

10

-1

10

0

10

1

10

2

10

3

10

4

10

5

10

6

10

7

10 -2 -3 10

10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

10 5

10 6

10 7

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Energy (eV)

Energy (eV)

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Conversion
10 4

232
10
3

Th

(barns)

10 2

10 1

capture
10
0

fission
10 -1

10 -2 -3 10

10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

10 5

10 6

10 7

10 4

Energy (eV)

233
10
3

U

(barns)

10 2

10 1

fission
10 0

10 -1

capture

10 -2 -3 10

10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

10 5

10 6

10 7

Energy (eV)

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Neutron Cycle in Reactor
E
2 MeV
Slowing down
p  PFNL  N 1
8

N 2    Pf PT NL fpPFNL  N 1

N2 N1

Leakage

  N1

PFNL   N 1

 n/fission

N2 k N1
ν ≈ 2.5
Pf  PT NL fpPFNL  N 1

Fast fission

Energy

Resonance abs.

Non-fissile abs.

Non-fuel abs.

1 eV

Fission 200 MeV/fission

PT NL  fpPFNL  N 1 f  pPFNL  N 1

Leakage
T8: Chain Reaction

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Fast Fission
10 4

235
10
3

U

(barns)

10 2

fission
10
1

10 0

capture
10
-1

10 -2 -3 10 10 4

10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

10 5

10 6

10 7

Energy (eV) 238
10
3

U

(barns)

10 2

capture
10 1

10 0

10 -1

fission
10
-2

10 -3

10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

10 5

10 6

10 7

Energy (eV)

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Simple Criticality Calculations

Number of fission neutrons due to both fast and thermal fission Number of fission neutrons due to only thermal fission p  Resonance escape probability
F a f F thermal utilization  a  Other a

N 2  N1   PFNL pfPTNL Pf

F f Pf  F  F a  a
F f

F f    Pf   F a
N2 k   fp PFNL PTNL N1 PFNL PTNL  1  k   fp

  1.65
k  1.04
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f  0.71 k  1.00

p  0.87 PFNL  0.97 PTNL  0.99
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Fuel Cycle (Thermal)
Fissile Materials

U nat

Fuel Factory

UE or U nat

Thermal Reactor

Actinides (MA)

Fiss. Pr. (FP)

DU
Spent Fuel

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Fuel Cycle (Fast)
239

Pu
Spent Fuel

Fuel Factory

Initial Loading
238

U

239

Pu

Fast Reactor

FP+MA

238

U

Spent Fuel

DU

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Neutron Economy

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Breeding Factor
B neutrons captured in fertile

1 neutron absorbed in fuel

η neutrons produced

1 neutron to maintain chain reaction

Non-fuel capture and losses C+L

  1  B  (C  L) B    1  (C  L)  1    2  (C  L)  2   2.2
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η Factor
η Isotope Thermal Fast (100%) (0.025eV) (>0.5 MeV)
235U 233U 239Pu

2.07 2.29 2.15

2.35 2.40 2.90

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Fission Neutrons
239Pu 235U

233U

239

  2.844  0.138  E   2.432  0.066  E
 2.349  0.150  E

235

E=1

233

  2.482  0.075  E
 2.412  0.136  E

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Conversion Factor
If B < 1 it is called conversion factor, C.
Number of 239Pu created per 1 neutron absorbed
238 a C  235   235 1  p  PFNL PT NL a

Reactor System BWR

Initial Fuel
235U

Conversion Cycle
238U

Conversion Ration 0.6

(2-4%)

→ 239Pu

PWR
PHWR (CANDU) HTGR LMFBR
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235U

(2-4%)
(5%)

238U
238U

→ 239Pu
→ 239Pu → 233U → 239Pu

0.6
0.8 0.8 1.0 – 1.6
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Unat
235U 239Pu

232Th 238U

(10-20%)

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Neutron Cycle in Reactor
E
2 MeV
Slowing down
p  PFNL  N 1
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N 2    Pf PT NL fpPFNL  N 1

N2 N1

Leakage

  N1

PFNL   N 1

 n/fission

N2 k N1
ν ≈ 2.5
Pf  PT NL fpPFNL  N 1

Fast fission

Energy

Resonance abs.

Non-fissile abs.

Non-fuel abs.

1 eV

Fission 200 MeV/fission

PT NL  fpPFNL  N 1 f  pPFNL  N 1

Leakage
T8: Chain Reaction

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Chain Reaction in Unat?
   
235 f 
235 f



235 c



238 c



 f235N 235

 f235N 235   c235N 235   c238N 238

 f235   c235

 f235  f235   235 238 238 235  c N N  f   c235   c238 1  e  e

579    2.42  0.547  1.32 579  101  2.72  139

Moderat or = H 2O  not possible (LWR) Moderat or = D2O  possible (P HWR) Moderat or = Graphit e  possible (RBMK)
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Doubling Time
Td
NF  [

for a breeder reactor is the amount of time required for the original fissile loading to double
at oms ] g

W Volume Rat ing R   f N  f    f  f  [ 3 ] cm W Mass Rat ing R   f N F  f  [ ] g M F is Mass of Fissile Fuel atoms  RC  M F N F  a    s    atoms  R P  BM F N F  a   s    R net  R P  RC   B  1 M F N F  a
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T d R net  M F N F

 f N F f Td   B  1 R  a
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Doubling Time for FR
 f N F f Td   B  1 R  a  f  3.36  1011 
J    fission  

atoms  N F  2.52  1021    g   a  2.15 barn ;  f  1.8 barn  MW  R  500   ton n e 239 P u   T d  22 yr

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Multiplication Factor
 1: N2  k    1: N1   1:

N(t)
subcritical critical supercritical

t

P (t ) is Production rate L(t ) is Loss rate P (t ) N (t ) k ; l  Lifetime of a neutron generation L(t ) L(t )  P (t )  dN (t ) k 1  P (t )  L(t )    1 L(t )  N (t ) dt l  L(t )   k 1  N (t )  N 0 exp  t  l 
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Chain Reaction by Fast N.
N (t )  N 0 e t ( k 1) l  N 0 e t T T l Reactor Period k 1   19 g cm3 ;  f  1.2 b;  t  6 b
235U

v( E  2 MeV )  2 109 cm s 19  6 1023  6 1024 t  N t  N A t   0.29cm 1 A 235  3.4 t  1 t  3.4cm    t   1.7  109 s v 2 109 f 1 9 8.5 109   l  5  8.5  10 s k  1.85  T   108 s t 5 0.85



N (t )  t   exp  8  N (0)  10 
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Time to Fission 1 kg 235U
1gatom  6 1023 atoms  235g 3 1024 3 10 can (theoretically) fission 235  1175g 23 6 10 10% leackage  1kg 235 U is fissioned by 3 1024 neutrons
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N(t)  t   3 1024  exp  8   t  5.6 107 s 1  10  Fission of 1 g of 1 kg of
235 235

U  1MWd  8.6 104 MJ  20 t TNT

U  20 kt TNT within 5.6 107 s

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Energy Comparison
TNT = Tri Nitro Toluol 1 kg of TNT = 4.2 MJ 1W = 3.1 1010 fission s Fission of 1 g of Fission of 1 kg of Bomb
235

U  8.6 10 4 MJ  20 t TNT U  20 kt TNT U is fissioned within 6 10-7 s  P=1.5 1014 MW

235

: 1 kg of

235

NPP (F-3): 1 kg of 235 U is fissioned within 8 hours  P=3 103MW F-3 consumes 3 bombs a day (as that droped on Nagasaki)

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Distribution of FP Mass

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Nuclide

Half-life (year)

Discharge from 3000 MWt LWR (kg/a) 14.5 4.5 166.0 76.7 25.4 15.5 16.6 3.0 0.6 0.2 0.1 13.4 13.4 23.2 7.3 1.0 5.8 9.4 31.8 0.4

Actinides
Np Pu 239 Pu 240 Pu 241 Pu 242 Pu 241 Am 243 Am 244 Cm
238 237

2 140 000 80 24 000 6 600 14 374 000 433 7 400 18
FISSION PRODUCTS

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Se Kr 90 Sr 93 Zr 99 Tc 107 Pd 126 Sn 129 I 135 Cs 137 Cs 151 Sm
85

79

65 000 11 28 1 500 000 210 000 6 500 000 100 000 15 700 000 3 000 000 30 T8: Chain Reaction 90

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Long-Lived Radio Isotopes

T8: Chain Reaction

Isotope

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Long-lived radioactive isotopes in LWR spent fuel
0,1 151Sm 137Cs 135Cs 129I 126Sn 107Pd 99Tc 93Zr 90Sr 85Kr 79Se 244Cm 243Am 241Am 242Pu 241Pu 240Pu 239Pu 238Pu 237Np 0,1 1 10 100 1 10 100

Discharge from 3000MWth LWR (kg/year)
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Half-Lives of L.L. Isotopes

T8: Chain Reaction

Isotope

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Half-life of long-lived isotopes in LWR spent fuel
151Sm 137Cs 135Cs 129I 126Sn 107Pd 99Tc 93Zr 90Sr 85Kr 79Se 244Cm 243Am 241Am 242Pu 241Pu 240Pu 239Pu 238Pu 237Np

10 10 10 10 10 10 10 10 Half-life (years)

0

1

2

3

4

5

6

7

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The END

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